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## Über dieses Buch

This book reports on recent achievements in stability and feedback stabilization of infinite systems. In particular emphasis is placed on second order partial differential equations, such as Euler-Bernoulli beam equations, which arise from vibration control of flexible robots arms and large space structures. Various control methods such as sensor feedback control and dynamic boundary control are applied to stabilize the equations.
Many new theorems and methods are included in the book. Proof procedures of existing theorems are simplified, and detailed proofs have been given to most theorems.
New results on semigroups and their stability are presented, and readers can learn several useful techniques for solving practical engineering problems.
Until now, the recently obtained research results included in this book were unavailable in one volume. This self-contained book is an invaluable source of information for all those who are familiar with some basic theorems of functional analysis.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The study of stability originates in mechanics. Early in the 17th century, a principle, called Torricelli’s principle[154]. was already in use: it says that if a system of interconnected heavy bodies is in equilibrium, the center of gravity is at the lowest point. This principle was applied to the study of general motion including, but not limited to. mechanical motion. In fact, any time process in nature can be thought of as motion, and to study stability is actually to study the effect of perturbations to motion.
Zheng-Hua Luo, Bao-Zhu Guo, Omer Morgul

### Chapter 2. Semigroups of Linear Operators

Abstract
Semigroups of linear operators are closely related to the solution of the following linear differential equation:
$$\dot u(t) = Au(t).u(0) = x \in X,$$
(2.1)
Zheng-Hua Luo, Bao-Zhu Guo, Omer Morgul

### Chapter 3. Stability of C0-Semigroups

Abstract
Stability analysis and feedback stabilization are issues of great importance in control system design. In this chapter, we study the stability of the following abstract Cauchy problem on a Danach space X:
$$\left\{ {\begin{array}{*{20}{c}} {\frac{{du(t)}}{{dt}} = Au(t).} \\ {u(0) = x \in X.} \end{array}} \right.$$
(2.1)
Zheng-Hua Luo, Bao-Zhu Guo, Omer Morgul

### Chapter 4. Static Sensor Feedback Stabilization of Euler-Bernoulli Beam Equations

Abstract
In this chapter, we consider stabilization problems of Euler-Bernoulli beam equations arising in the area of space and industrial robots with lightweight and flexible arms, as well as in the area of flexible space structures. We shall first derive a general model for a Euler-Bernoulli beam with a rigid tip body. The model is more general than those models in the existing literature in the sense that both bending and torsional vibrations of the beam will be considered, and the tip body is allowed to be a rigid body. Although the derived model looks complicated, it can be reformulated into a simple abstract equation in some appropriately defined Hilbert spaces. For this reason, in the sections which follow, we shall consider a simplified version of this model by assuming a free beam (with no tip bodies) without loss of generality.
Zheng-Hua Luo, Bao-Zhu Guo, Omer Morgul

### Chapter 5. Dynamic Boundary Control of Vibration Systems Based on Passivity

Abstract
In this chapter, we consider feedback stabilization of a class of passive infinite dimensional systems by means of dynamic boundary control. The notion of passivity was developed in connection with circuit theory in the late ′50s where the basic motivation was to investigate the behavior of circuits composed of passive circuit elements such as resistors, capacitors and inductors, see [61]. This concept was then introduced into control systems, see [2], [50], [156], [164]. To motivate the concept of passivity, let us consider the following situation: Let S be a dynamical system with an input vector u= (u1,…,um)TRmand an output vector y = (y1,…,ym)TRm. Let H be the Hilbert space in which the solutions of S evolves, and let E(t): HR be a positive time function which depends on the solutions of S. Assume that the time derivative of E(t) along the solutions of S satisfies
$$\dot E(t) = {u^T}y = \sum\limits_{i = 1}^m {{u_i}{y_i}.}$$
(5.1)
Zheng-Hua Luo, Bao-Zhu Guo, Omer Morgul

### Chapter 6. Other Applications

Abstract
In Chapter 4, we applied the energy multiplier method and frequency domain criteria to determine the exponential stability of linear dynamic systems which generate C0-semigroups on Danach or Hilbert spaces. For systems which satisfy the spectrum-determined growth condition, the exponential stability can be examined by analyzing the spectrum distribution of the systems. One advantage of this method over others is that we are able to know not only whether the systems are exponentially stable, but also the exponential decay rate.
Zheng-Hua Luo, Bao-Zhu Guo, Omer Morgul

### Backmatter

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